Square Root Calculator › Square root of 392
Square Root of 392
√392 ≈ 19.79899
392 is not a perfect square, so √392 is an irrational number that is approximately 19.79899. In simplified radical form, √392 = 14√2.
√392 lies between √361 = 19 and √400 = 20.
| Number | 392 |
| Square root (decimal) | 19.79899 |
| Simplified radical | 14√2 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 392
392 = 2^3 × 7^2. This is what determines the simplified radical: for √392 you pull out one copy of any prime that appears twice.
Here the paired factors 2 and 7 come out from under the root, leaving √392 = 14√2.
Estimating √392 by hand
√392 falls between √361 = 19 and √400 = 20, so it starts at 19.something. The Babylonian method refines that guess by averaging it with 392 ÷ the guess:
| Step | Estimate |
| Start | 19 |
| Step 1 | 19.815789 |
| Step 2 | 19.798997 |
| Step 3 | 19.79899 |
After 3 steps the estimate is 19.79899, matching √392 ≈ 19.79899.
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Frequently asked questions
Is 392 a perfect square?
No, 392 is not a perfect square, so its square root is irrational and approximately 19.79899.
What is the square root of 392 in simplest radical form?
√392 = 14√2 in simplest radical form.